I suggest for a 9/1 rental start looking either mid or latter July. I wouldn't wait until August, by then everyone else will be looking and the school season will be around the corner. If you can lock in (security deposit required with a possible earlier occupancy date) something sooner, even better.
Rentals are really hot and I have seen multiple applications and offers of up to 3 months advance payment to secure and win the apartment. Amazing, isn't it!?